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Policy gradients methods often achieve better performance when the change inpolicy is limited to a small Kullback-Leibler divergence. We derive policygradients where the change in policy is limited to a small Wasserstein distance(or trust region). This is done in the discrete and continuous multi-armedbandit settings with entropy regularisation. We show that in the small stepslimit with respect to the Wasserstein distance $W_2$, policy dynamics aregoverned by the Fokker-Planck (heat) equation, following theJordan-Kinderlehrer-Otto result. This means that policies undergo diffusion andadvection, concentrating near actions with high reward. This helps elucidatethe nature of convergence in the probability matching setup, and providesjustification for empirical practices such as Gaussian policy priors andadditive gradient noise.

On Wasserstein Reinforcement Learning and the Fokker-Planck equation